"Tomorrow is Yesterday" showed that the ship was pretty banged up by the slingshot and barely came through it intact.

Or then it could be argued that both this and ST4 featured the exact same risk: the crew loses consciousness and therefore risks letting the ship get wrecked (say, by slamming into an atmosphere), but damage as such is not inevitable at all.

Once the risks are known, preparing for the loss of consciousness and taking precautions regarding the unsupervised arrival at the destination is easy enough, allowing our heroes to conduct the "Assignment: Earth" mission without damage.

It just seems like post TOS, unless using the exact means they used to get there (Borg tachyon hole, the wormhole in "Yesterday's Enterprise") then they're stuck.

I'm not sure I can recognize any good examples of this. In "Yesterday's Enterprise", the heroes were starved of means to do anything much; flying the E-C to a star, let alone completing a slingshot maneuver and then somehow returning to Narendra III, seemed well beyond their capabilities. In all other TNG time travel stories, the heroes were up the timecreek not just without a paddle, but without a canoe as well: you can't slingshot if you don't have a starship!

When time travel took place in a way that involved a starship, there wasn't really an incentive to escape from the situation by slingshotting: the heroes had a task to accomplish first. Say, in "Trials and Tribble-ations", they have to hunt down Arne Darvin before they can go home. Also, the means of getting into the past is usually also the logistically most convenient for subsequent travel in that situation. In "Past Tense", slingshotting with the Defiant to locate the missing people would have taken much more effort (and time!) than using the means-of-the-week, especially as they had to do it repeatedly - and were chasing after somebody who had used the means-of-the-week in the first place, so following the spoor made perfect sense.

There are plenty of occasions where time travel would have made sense as a means of "rewinding the clock" to compensate for the fact that the heroes arrived too late. But using it to compensate for time travel that had already occurred was generally not tactically necessary or desirable.

Picard's line means that it can be done is probably public knowledge given there were hundreds of Enterprise crew, so it can't be "Captain's eyes only"

Swearing the hundreds of Kirk's crew to silence would keep it from becoming public knowledge for all eternity afterwards, though. And the exchange between Picard and Riker certainly didn't indicate that others aboard Picard's ship would be in the know: rank hath its privileges.